What type of wordplay joins two phrases together on a single syllable?

Solution 1:

Your example involves enjambment, homophones, and mind rhyme.

Enjambment is the running together of lines. In a normal case of enjambment, a complete clause is separated over two lines, as in this couplet from Auden:

Keep running if you want to reach
the point of knowing where you stand.

But with your example of

I don't want my breakfast, because it tastes like—
Shih Tzus make good housepets, they're cuddly and sweet,
Monkeys aren't good to have, because they like to beat their—
Meeting in the office...

each line is interrupted and a new one began. But the initial sounds of the subsequent lines can be viewed as completing the previous lines. This is because these initial sounds are homophones for what you'd expect the previous line to end with. Homophones are words that are pronounced the same but differ in meaning (for example meet and meat). The homophonic material starting the subsequent line can thus be viewed as the enjambed completion of the previous line.

As the lines are interrupted, their actual completions are only suggested. The device of suggesting a rhyme without actually giving it is called mind rhyme.

In fact, your specific example is mentioned in the Wikipedia article for mind rhyme (which also mentions an earlier example, Miss Susie).

Mind rhyme that involves sexual innuendo or obscene language is sometimes called teasing rhyme, although this term is also sometimes used more broadly to mean any mind rhyme.

Very generally, the song employs many puns, where a pun is "a joke exploiting the... the fact that there are words that sound alike but have different meanings."